Describe the principle of virtual work, and illustrate your description by an example. \(ABCDE\) is a framework of four equal rods, each of weight \(w\) and length \(l\), freely hinged at \(B, C\) and \(D\). The ends \(A\) and \(E\) are freely hinged at two fixed points on the same horizontal level, and the points \(B\) and \(D\) are connected by a string. The whole framework hangs symmetrically below \(AE\) in the shape of a W, and \(AB, BC\) make acute angles \(\theta\) and \(\phi\) respectively with the vertical. A weight \(W\) is suspended from \(C\). Using the principle of virtual work, or otherwise, prove that the tension in the string is \[ \frac{3w+W}{2} \tan\theta + \frac{w+W}{2} \tan\phi. \]
A variable tangent to a conic S meets two fixed perpendicular tangents a, b at P, Q respectively, and the perpendiculars to a, b at P, Q meet at L. Prove that, if S is a central conic, the locus of L is a rectangular hyperbola whose asymptotes are the tangents of S parallel to a, b; and that this hyperbola passes through the points of contact of a and b with S. What is the locus of L when S is a parabola?
If the roots \(x_1, x_2, x_3\) of the equation \[ x^3 = 3p^2x + q \] are all real and distinct, prove that \(4p^6 > q^2\). Obtain the equation whose roots are \(x_1^2 - r, x_2^2 - r, x_3^2 - r\), where \(3r = x_1^2 + x_2^2 + x_3^2\). Given \(x_1^2 < x_2^2 < x_3^2\), prove that \[ x_2^2 - x_1^2 < x_3^2 - x_2^2 \] if and only if \(2p^6 < q^2\).
Shew that if a focus be taken as pole, then the polar equation to a conic may be written in the form \[ \frac{l}{r} = 1 + e \cos \theta. \] A circle through the focus meets the conic in four points whose distances from the focus are \(r_1, r_2, r_3, r_4\). Prove that \[ \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \frac{1}{r_4} = \frac{2}{l}. \]
Find the centre of mass of a thin uniform wire of length \(l\) bent into an arc of a circle of radius \(a\). \(ABC\) is a uniform semi-circular wire, of weight \(w\) per unit length, and rests in a vertical plane with \(AC\) horizontal and \(B\) in contact with the ground. Find expressions for the bending moment and the shearing force at any point of the wire.
If \(\alpha = 2\pi/7\), prove that \[ \sin\alpha + \sin 2\alpha + \sin 4\alpha = \tfrac{1}{2}\sqrt{7}. \]
A card is drawn at random from an ordinary pack and is then replaced. A second card is then drawn at random and afterwards replaced, then a third, and so on. Prove that the chances in favour of all the four suits having turned up during the first seven draws are very slightly better than 21 out of 41.
If \(\alpha, \beta, \gamma\) are the eccentric angles of three points \(P, Q, R\) on an ellipse, the normals at which are concurrent, prove that \[ \sin(\beta+\gamma) + \sin(\gamma+\alpha) + \sin(\alpha+\beta) = 0. \] \(A\) is a vertex of the ellipse, and \(P', Q', R'\) are points on the ellipse such that \(AP'\) is parallel to \(QR\), \(AQ'\) parallel to \(RP\), \(AR'\) parallel to \(PQ\). Prove that the centre of gravity of the triangle \(P'Q'R'\) lies on the major axis.
A projectile is fired in a fixed vertical plane with maximum velocity \(u\). Shew that all points which it can reach lie within or on a parabola, which is the envelope of those trajectories for which the velocity of projection is \(u\). Obtain its equation and shew that the point of projection is a focus of the parabola. \(A\) and \(B\) are two similar guns each capable of firing a shell with muzzle velocity 1000 feet per second. \(A\) is at the top of a tower 100 feet high and \(B\) is on the ground. Find the area of ground in which \(B\) may be placed so that \(B\) may be hit by \(A\) but \(A\) may not be hit by \(B\).
Prove that if \[ \begin{vmatrix} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{vmatrix} = 0, \] there exist three numbers \(x, y, z\), not all 0, satisfying simultaneously the three equations \begin{align*} ax + by + cz &= 0, \\ a'x + b'y + c'z &= 0, \\ a''x + b''y + c''z &= 0. \end{align*} Hence (or otherwise) prove that if \[ |a| > |b|+|c|, \quad |b'| > |c'|+|a'|, \quad |c''| > |a''|+|b''|, \] the above determinant cannot vanish. [\(|a|\) denotes the absolute value of \(a\).]